
Since \(|D_n| \le 2\Vert g \Vert _{\infty }\) , applying [1]} with expectation conditional to \(Z_0, \ldots , Z_{n - 1}\) , \(a = -2\Vert g \Vert _{\infty }\)  and \(b = 2\Vert g \Vert _{\infty }\)  gives us
\(\mathbb {E}_z \left[ \mathrm {e}^{\theta D_n} \mid Z_0, \ldots , Z_{n - 1} \right] \le \mathrm {e}^{2\theta ^2 \Vert g \Vert _{\infty }^2}.\) 
